Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pwsvscafval | Structured version Visualization version GIF version |
Description: Scalar multiplication in a structure power is pointwise. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsvscaval.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsvscaval.b | ⊢ 𝐵 = (Base‘𝑌) |
pwsvscaval.s | ⊢ · = ( ·𝑠 ‘𝑅) |
pwsvscaval.t | ⊢ ∙ = ( ·𝑠 ‘𝑌) |
pwsvscaval.f | ⊢ 𝐹 = (Scalar‘𝑅) |
pwsvscaval.k | ⊢ 𝐾 = (Base‘𝐹) |
pwsvscaval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
pwsvscaval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
pwsvscaval.a | ⊢ (𝜑 → 𝐴 ∈ 𝐾) |
pwsvscaval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
pwsvscafval | ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsvscaval.t | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝑌) | |
2 | pwsvscaval.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
3 | pwsvscaval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
4 | pwsvscaval.y | . . . . . . 7 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
5 | pwsvscaval.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑅) | |
6 | 4, 5 | pwsval 16146 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
7 | 2, 3, 6 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
8 | 7 | fveq2d 6195 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
9 | 1, 8 | syl5eq 2668 | . . 3 ⊢ (𝜑 → ∙ = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))) |
10 | 9 | oveqd 6667 | . 2 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋)) |
11 | eqid 2622 | . . 3 ⊢ (𝐹Xs(𝐼 × {𝑅})) = (𝐹Xs(𝐼 × {𝑅})) | |
12 | eqid 2622 | . . 3 ⊢ (Base‘(𝐹Xs(𝐼 × {𝑅}))) = (Base‘(𝐹Xs(𝐼 × {𝑅}))) | |
13 | eqid 2622 | . . 3 ⊢ ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) = ( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅}))) | |
14 | pwsvscaval.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
15 | fvex 6201 | . . . . 5 ⊢ (Scalar‘𝑅) ∈ V | |
16 | 5, 15 | eqeltri 2697 | . . . 4 ⊢ 𝐹 ∈ V |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
18 | fnconstg 6093 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝐼 × {𝑅}) Fn 𝐼) | |
19 | 2, 18 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼 × {𝑅}) Fn 𝐼) |
20 | pwsvscaval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐾) | |
21 | pwsvscaval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
22 | pwsvscaval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
23 | 7 | fveq2d 6195 | . . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
24 | 22, 23 | syl5eq 2668 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
25 | 21, 24 | eleqtrd 2703 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐹Xs(𝐼 × {𝑅})))) |
26 | 11, 12, 13, 14, 17, 3, 19, 20, 25 | prdsvscaval 16139 | . 2 ⊢ (𝜑 → (𝐴( ·𝑠 ‘(𝐹Xs(𝐼 × {𝑅})))𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)))) |
27 | fvconst2g 6467 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) | |
28 | 2, 27 | sylan 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐼 × {𝑅})‘𝑥) = 𝑅) |
29 | 28 | fveq2d 6195 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = ( ·𝑠 ‘𝑅)) |
30 | pwsvscaval.s | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑅) | |
31 | 29, 30 | syl6eqr 2674 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥)) = · ) |
32 | 31 | oveqd 6667 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥)) = (𝐴 · (𝑋‘𝑥))) |
33 | 32 | mpteq2dva 4744 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
34 | 20 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ 𝐾) |
35 | fvexd 6203 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑋‘𝑥) ∈ V) | |
36 | fconstmpt 5163 | . . . . 5 ⊢ (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴) | |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐼 × {𝐴}) = (𝑥 ∈ 𝐼 ↦ 𝐴)) |
38 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
39 | 4, 38, 22, 2, 3, 21 | pwselbas 16149 | . . . . 5 ⊢ (𝜑 → 𝑋:𝐼⟶(Base‘𝑅)) |
40 | 39 | feqmptd 6249 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐼 ↦ (𝑋‘𝑥))) |
41 | 3, 34, 35, 37, 40 | offval2 6914 | . . 3 ⊢ (𝜑 → ((𝐼 × {𝐴}) ∘𝑓 · 𝑋) = (𝑥 ∈ 𝐼 ↦ (𝐴 · (𝑋‘𝑥)))) |
42 | 33, 41 | eqtr4d 2659 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐴( ·𝑠 ‘((𝐼 × {𝑅})‘𝑥))(𝑋‘𝑥))) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
43 | 10, 26, 42 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 ↦ cmpt 4729 × cxp 5112 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 Xscprds 16106 ↑s cpws 16107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-prds 16108 df-pws 16110 |
This theorem is referenced by: pwsvscaval 16155 pwsdiaglmhm 19057 pwssplit3 19061 frlmvscafval 20109 |
Copyright terms: Public domain | W3C validator |